Real-rooted atlas

A038333 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(28 + 19\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 784 + 1064\,x + 361\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
078410643612209
12195244688303246859103823
2614656166835216981447682081303214879681
317210368583923207924672053774560182449402476099229345007
448189030419619819523328362240301137536015325749604159846324704588110779215329
51349292851264091410432130471799808147557392640100128230720407664939369220992676893871739506623120463
63778019983362050925133824487094719283266105711902725607180920320304389821388810327511797122002272695361698356304123811286661761
71057845595340864604141715456175354098941952277643989991424282601918384128191765587474944867510990958082522863596153642798578863323226876977791119130473102767
8296196766695424200990663114752061373934629683201110575959965696013188089524592640107388728985968646072576936706560235467268974336059918010408648090352555378120613106625780152599132235830049
9829350946747187261905124239343616210035242954915840427571744586792960580275939082076160551262142127972352374070739301124096181309797110238720615158240195452801391429352823048018883684074027081164902588982192472159215084012303
1023221826508921241618909201585835868167057184163285172224159626784645736038402437158944144719872026460582822142672896209479614008629493761218401836580804198451673292176418035201558400875161813760317245892443654944391407269898015842213314919066161116191483108948578241
1165021114224979476485735791147703546675223352863958507297177658104149611047917977698569539518742003712012039565184074916167681089293992844873367552739163780859021213696376181567044323296256141814479639725052160384925016164967998727123612312143888288805646630540082604420529834622570595460999706120583177327
121820591198299425341441729561638384454074368762860222644571707801620706206043209803497472386392594913468654551045243899502397074597478453375405649398795010048413931717281051879669762457719571356245535539211118255203754444089344377226515841668638745693081867545346806963215790673958585619038416484769517204767128799006685782884121256666986187667409334369
1350976553552383909560325188684915153362223104024646253346978470559744072471721151234312241152014753171805786984991948802202437791006771330940928249085226363861043380224021731415157255223682662401474631742813747321323520778277864262811086254080316870273307001656546304977359609226141473113602210694354201986665376034618015986130010968803355828080288113308201518112702987479829912063348350820368238715343
141427343499466749467688961549687227992470850633728788680107103311057911808024974870058271516833873920550785080749380773032755208969928457918486875104870411159018141100974743434035210817415544944822455369728082579377597569849994117120498097833128199095202611202365964707358945702212403287571420986662275990978562475977676706225065221120516962372059541497134080751705489984537381036806801144909383909637952288441413567621167681566977372488557307219621121
Technical details
Node id
n-a4164ed904ec2910 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-304eeb06eba1cf88

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              784,              1064,                361;
  1:            21952,             44688,              30324,               6859;
  2:           614656,           1668352,            1698144,             768208,             130321;
  3:         17210368,          58392320,           79246720,           53774560,           18244940,            2476099;
  4:        481890304,        1961981952,         3328362240,         3011375360,         1532574960,          415984632,           47045881;
  5:      13492928512,       64091410432,       130471799808,       147557392640,       100128230720,        40766493936,         9220992676,          893871739;
  6:     377801998336,     2050925133824,      4870947192832,      6610571190272,      5607180920320,      3043898213888,      1032751179712,       200227269536,       16983563041;
  7:   10578455953408,    64604141715456,    175354098941952,    277643989991424,    282601918384128,    191765587474944,     86751099095808,     25228635961536,     4279857886332,      322687697779;
  8:  296196766695424,  2009906631147520,   6137393462968320,  11105759599656960,  13188089524592640,  10738872898596864,   6072576936706560,   2354672689743360,   599180104086480,    90352555378120,    6131066257801;
  9: 8293509467471872, 61905124239343616, 210035242954915840, 427571744586792960, 580275939082076160, 551262142127972352, 374070739301124096, 181309797110238720, 61515824019545280, 13914293528230480, 1888368407402708, 116490258898219;
%F Formula
The row polynomials satisfy P(n,x) = (28+19*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 1064*x + 361*x^2.
%t Mathematica
P[0, x_] := 784 + 1064*x + 361*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 19*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 784 + 1064*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 19*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.